Abstract
For conservative dynamical systems having two degrees of freedom Birkhoff has established the existence of two classes of periodic orbits. The first consists of stable-unstable pairs close to periodic orbits of the stable type, and the second of orbits having fixed points (in a suitable surface of section) close to homoclinic points. In this paper orbits of the latter type are listed, and their evolution followed as a function of the energy. For the energy at which they were first computed, all were unstable; but they evolved, with diminishing energy, into one orbit of the stable type which appears to be a member of the first class of orbits mentioned above.
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References
Birkhoff, G. D.: 1933,Dynamical Systems, published by the American Math. Soc., see Chapter 8, Section 2. Subsequent proofs, such as that by Birkhoff, G. D. and Lewis, D. C.: 1933,Annali di Matematica Pura ed Applicata, series 4,12, 117 are concerned with systems of higher degrees of freedom.
Birkhoff, G. D.: 1927,Acta Math. 50, 333.
Danby, J. M. A.: 1970, in G. E. O. Giacaglia (ed.),Periodic Orbits, Stability and Resonances, D. Reidel Publ. Co., Dordrecht-Holland, p. 272.
Deprit, A. and Henrard, J.: 1967, ‘A Manifold of Periodic Orbits’,Advances in Astronomy and Astrophysics, Academic Press.
Poincaré, H.: 1957,Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 1, Dover. The basic steps are described very clearly in Pars, L. A.: 1965,A Treatise on Analytical Dynamics, Chapter 30, Heinemann.
Walker, G. H. and Ford, J.: 1969,Phys. Rev. 188, 416.
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Danby, J.M.A. The evolution of periodic orbits close to homoclinic points. Celestial Mechanics 8, 273–280 (1973). https://doi.org/10.1007/BF01231428
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DOI: https://doi.org/10.1007/BF01231428